
Initialization:

Initial distribution function \(f(x_1, x_2, v_1, v_2, t=0)\)  in a low rank format (REF )-(REF ).


For each time step evolution from \(t^n\)  to \(t^{n+1}\) : update \({\bf f}^{n+1}\)  from \({\bf f}^n\)  in the low rank format.

Compute the charge density \(\rho ^{n}\)  in the \((x_1, x_2)\)  full grid format, followed by computing \({\bf E}^n = (E_1^n, E_2^n)\)  from the Poisson equation (REF ) by fast Fourier transform or a high order finite difference algorithm.

Update \({\bf f}^{n+1,*}\) , by adding basis according to the dimension tree \(\mathcal {T}\)  as shown in Figure REF . The procedure is similar to that outlined in [1]}.

Compute \({\rho }^{n+1, *}\) , \({\bf J}_1^{n+1, *}\) , \({\bf J}_2^{n+1, *}\) , \({\kappa }^{n+1, *}\)  by low rank numerical integration in velocity, i.e. (REF ). Compute \({\bf f}_1\)  from (REF ) with \(\rho ^{n+1, *}\) , \({\bf J}_1^{n+1, *}\) , \({\bf J}_2^{n+1, *}\) , \(\kappa ^{n+1, *}\) .

Compute \({\bf f}_2\doteq {\bf f} - {\bf f}_1\)  and perform a weighted SVD truncation on \({\bf f}_2\)  in the low rank 2D2V format [2]} to obtain
\(\left( \sqrt{\bf w} \star \mathcal {T}_\varepsilon (\frac{1}{\sqrt{\bf w}} \star {\bf f}_2)\right).\) 
Finally, we apply the \((I-P_\mathcal {N})\)  operator to \(\left( \sqrt{\bf w} \star \mathcal {T}_\varepsilon (\frac{1}{\sqrt{\bf w}} \star {\bf f}_2)\right)\)  to ensure its zero mass, momentum and kinetic energy after truncation.

Compute \({\bf f}^M_1\) .

Update macroscopic mass, momentum and energy density, \(\rho ^{n+1, M}\) , \({\bf J}_1^{n+1, M}\) , \({\bf J}_2^{n+1, M}\) , \({e}^{n+1, M}\) , using the kinetic flux vector splitting, in a flux-difference form using the same second order SSP multi-step method in Step 2(b).

Compute \({\bf E}^{n+1}\)  from \(\rho ^{n+1, M}\)  by Poisson solver as in Step 2(a).

Compute \(\kappa ^{n+1, M}\)  by subtracting energy from the electrostatic field from \({e}^{n+1, M}\) .

Construct \({\bf f}^M_1\)  from (REF ), but with \(\rho ^{n+1, M}\) , \({\bf J}_1^{n+1, M}\) , \({\bf J}_2^{n+1, M}\) , \(\kappa ^{n+1, M}\) .


Update the compressed low-rank solution via (REF ),
\({\bf f}^{n+1} \doteq T^M_c({\bf f}) = {\bf f}^M_1 +(I-P_\mathcal {N})\left(\sqrt{\bf w} \star \mathcal {T}_\varepsilon (\frac{1}{\sqrt{\bf w}} \star {\bf f}_2)\right).\) 


